Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials

We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 684} (2010), 173-176]. These $X_{\ell}$ polynomials are global solutions of second order Fuchsian differential equations with $\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape invariance, the forward and backward shift operations, invariant polynomial subspaces under the Fuchsian differential operators, the Gram-Schmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the $X_{\ell}$ polynomials.


Introduction
Four sets of infinitely many exceptional (X ℓ ) polynomials satisfying second order differential equations were introduced recently by two of the present authors [1,2]. They were obtained as the main part of the eigenfunctions of exactly solvable one-dimensional quantum mechanical systems which were deformations of the well-known solvable systems of the radial oscillator [3,4] and the trigonometric Darboux-Pöschl-Teller (DPT) potential [5] by a degree ℓ eigenpolynomial. Thus the orthogonality and completeness of the X ℓ polynomials are automatically guaranteed. These polynomials, termed exceptional Laguerre and Jacobi polynomials, have two types in each family, L1, L2 and J1 and J2. The Laguerre family L1 (L2) is obtained from the Jacobi family J1 (J2) by the well-known limit (E.28), which takes the Jacobi polynomials to the Laguerre polynomials. The J1 and J2 are mirror images of each other, see (E.14), but their limits L1 and L2 are clearly distinct. These polynomials are exceptional in the sense that they start at degree ℓ (ℓ = 1, 2, . . .) rather than degree 0 constant term. Thus they are not constrained by Bochner's theorem [6], which states that the orthogonal polynomials (starting with degree 0) satisfying a second order differential equations are very limited. Namely, they are only the classical orthogonal polynomials, the Hermite, Laguerre, Jacobi and Bessel polynomials.
The concept of exceptional orthogonal polynomials was introduced in 2008 by Gomez-Ullate et al. [7,8]. Within the Sturm-Liouville theory they constructed X 1 Laguerre and X 1 Ja-cobi polynomials, which turned out to be the first members of the infinite families. The results in [7,8] were reformulated in the framework of quantum mechanics and shape-invariant potentials [9] by Quesne and collaborators [10,11]. They found the first member of the deformed hyperbolic DPT potential family, which was also given in [1]. Quantum mechanical reformulation offers two merits. Firstly, the orthogonality and completeness of the obtained eigenfunctions are guaranteed. Secondly, the well established solution mechanism of shape invariance combined with Crum's method [12], or the so-called factorisation method [3] or the susy quantum mechanics [4] is available. For systems consisting of discrete eigenvalues only, shape invariance is a well-known sufficient condition for exact solvability of one-dimensional Schrödinger equation. The discovery of the four sets of infinitely many exceptional orthogonal polynomials was achieved by pursuing shape invariant deformation [1,2]. To be more precise, deformation of the potential by keeping the property of shape invariance. After the first paper on infinitely many X ℓ polynomials [1], Quesne reported a type II X 2 Laguerre polynomials [13,14], which led to the discovery of the L2 family of X ℓ Laguerre polynomials [2]. In a previous paper [15], two of the present authors unveiled infinitely many polynomial identities of degree 3ℓ involving cubic products of the Laguerre or the Jacobi polynomials, which encode the information of exact solvability of the differential equations governing the X ℓ polynomials.
In this paper we explore various properties of the X ℓ polynomials. We emphasise that the J1 and J2 polynomials are the global solutions of a Fuchsian differential equation having ℓ+3 regular singularities. They are located at ±1, ∞ and the ℓ zeros of the polynomial ξ ℓ (η; λ) (2.2), (2.4) which is used for the deformation. Factorisation and shape invariance are reformulated accordingly, leading to Rodrigues formulas and the forward and backward shift operations. The existence of the extra regular singularities implies that the ordinary vector space spanned by degree n polynomials V 0,n = Span [1, x, . . . , x n ] is not invariant under the Fuchsian differential operator (6.2). Appropriate invariant polynomial subspaces are introduced and their properties are used to derive the explicit forms of the exceptional polynomials. Some of the important subjects in orthogonal polynomial theory, namely, the Gram-Schmidt orthonormalisation, the generating functions, three term recurrence relations, the zeros of these orthogonal polynomials, etc, are also discussed. New infinitely many polynomial identities underlying the forward and backward shift operations are also reported.
The plan of this paper is as follows. In Section 2 we present the explicit forms of the four sets of infinitely many exceptional orthogonal polynomials together with their weight functions and the normalisation constants. They are equal to those reported earlier [1,2], but look much simpler than the original ones. The new forms of the polynomials reveal the structure of the theory. In Section 3 the Fuchsian differential equations governing these polynomials are discussed. Shape invariance and Rodrigues formulas are presented in Section 4. The identities underlying the forward and backward shift relations are presented in Section 5. The polynomial subspaces invariant under the Fuchsian differential operator are discussed in Section 6. This gives another concise proof of the new forms of the exceptional polynomials. In Section 7 we provide the integration formula which is essential for the Gram-Schmidt construction in Section 8. Section 9 gives the generating functions for the X ℓ polynomials. The double generating function, that is, the generating function of the generating functions, is presented for the L1 and L2 exceptional Laguerre polynomials. A substitute of the three term recurrence relations for the X ℓ polynomials is introduced in Section 10. In Section 11 we state the qualitative features of the extra zeros of the X ℓ polynomials without proof. The final section is for a summary and comments. Some technical details are relegated to Appendices. The equality of the new and original forms of the X ℓ polynomials is demonstrated in Appendix A. Forward and backward shift relations are proved in Appendix B. Derivation of the integration formula is provided in Appendix C. The properties of the X ℓ Jacobi polynomials as the solutions of the quantum mechanical systems with deformed hyperbolic DPT potentials are summarised in Appendix D. A concise summary of some important properties of the Laguerre and Jacobi polynomials is given in Appendix E for self-containedness.
Throughout this paper we stick to the notation of our previous papers [1,2,15,16]. Reference to the quantum mechanical language is made minimal in order to attain wider readership than before. Most concepts and formulas are common to the four sets of exceptional polynomials. As far as possible we use generic formulas valid for all the four different sets of X ℓ polynomials, in order to emphasise the underlying structure and at the same time to avoid redundancy.

Exceptional Laguerre and Jacobi polynomials
Here we present four sets of infinitely many exceptional orthogonal polynomials [1,2], among them two are deformations of the Laguerre polynomials, and the others are deformations of the Jacobi polynomials. They are expressed as a bilinear form of the original polynomials, the Laguerre or Jacobi polynomials and the deforming polynomials, depending on the set of parameters λ and their shifts δ and a non-negative integer ℓ, which is the degree of the deforming polynomials. It is important to stress that these explicit forms (2.1)-(2.4) can be derived by the Darboux-Crum transformations starting from the original (Laguerre or Jacobi) polynomials [17,18]. The two sets of exceptional Laguerre polynomials (ℓ = 0, 1, 2, . . ., n = 0, 1, 2, . . .) are: in which λ def = g > 0 and δ def = 1 and The two sets of exceptional Jacobi polynomials (ℓ = 0, 1, 2, . . ., n = 0, 1, 2, . . .) are 1 : (J2) It should be remarked that the naming J1 and J2 are interchanged from the previous ones [2] in order to respect the logical consistency rather than hysteresis.

Fuchsian dif ferential equations with extra ℓ regular singularities
The exceptional Laguerre and Jacobi polynomials satisfy a second order linear differential equation in the entire complex η plane: in which the eigenvalue E n is defined in (2.19). The X ℓ polynomials are not constrained by Bochner's theorem [6] by the very fact that they start at degree ℓ (2.9) instead of degree 0 constant term. As with the Laguerre and Jacobi differential equations (2.17), the second order differential operator H ℓ (λ) is factorised into the product of the forward shift operator F ℓ (λ) and the backward shift operator B ℓ (λ): The differential operator H ℓ (λ) is obtained from the factorised quantum mechanical Hamiltonian (16) of [1] and (3) of [2], by similarity transformation. Note that c 1 (η,λ+ℓδ) It is straightforward to derive the explicit form of H ℓ (λ): Use is made of the second order differential equations for ξ ℓ (η; λ + δ) (2.18) and the identity (2.20) to derive the above simple result. For ℓ = 0 the above differential equation (3.1) with (3.5) reduces to the second order differential equation for the Laguerre or Jacobi polynomials: which has, as is well-known, one regular singularity at η = 0 and one irregular singularity at η = ∞ for the Laguerre case and three regular singularities at η = ±1, ∞ for the Jacobi case. For a non-negative integer ℓ, the singularity structure of the second order differential equation (3.1) is again quite simple. It has extra regular singularities at the ℓ zeros of the deforming polynomial ξ ℓ (η; λ): and the corresponding exponents are the same for all the singular points: In other words, the alternative solution of the second order linear differential equation (3.1) has a cubic zero at η = η j . It is singular at η = 0, ∞ for the L1 and L2 and at η = ±1 for the J1 and J2 with the exponents replaced by g → g + ℓ, h → h + ℓ. To the best of our knowledge, the two sets of X ℓ orthogonal polynomials, J1 and J2 are the first examples of global solutions of Fuchsian differential equations having as many as ℓ + 3 regular singularities and forming a complete orthogonal system. The L1 and L2 are confluent types obtained from J1 and J2 by certain limits (E.28). By the way, it is elementary to show that ξ ℓ (η; λ) has only simple zeros. Let us write down the explicit form of the above differential equation for the four cases: Let us remark that the zeros of the shifted deforming polynomial ξ ℓ (η; λ + δ) are regular points.
It is straightforward to verify by direct calculation that the L1, L2, J1 and J2 X ℓ polynomials (2.1)-(2.4) for lower ℓ and n really satisfy the above differential equations (3.9)-(3.12). For analytical proof see the subsequent sections.
This much is a trivial consequence of the factorisation of H ℓ (λ) (3.2). The essential property of the two associated differential operators H ℓ (λ) and H ℓ (λ) has the same shape as H ℓ (λ) with shifted parameters, λ → λ + δ, and an additive constant corresponding to the lowest eigenvalue measured from the bottom: The system is called shape invariant if the above condition is satisfied. See [1,2,15] for the definition of shape invariance within the framework of quantum mechanics. The above is an equivalent definition within the framework of the Sturm-Liouville theory. Shape invariance allows us to choose the normalisation of {P (1) ℓ,n (η; λ)} to achieve Then the above two relations (4.1), the forward and backward shift relations, give the constraints on the functional forms of {P ℓ,n (η; λ)}: where f n (λ) and b n−1 (λ) are given in (2.25). These amount to a version of Rodrigues formula expressing P ℓ,n (η; λ) in terms of repeated application of the backward shift operators on the lowest degree eigenpolynomials with the n-th shifted parameters, P ℓ,0 (η; λ + nδ) = ξ ℓ (η; λ + (n + 1)δ): where n−1 k=0 a k = a 0 a 1 · · · a n−1 . For the X ℓ Laguerre and Jacobi polynomials, the Rodrigues formula reads explicitly: For ℓ = 0, ξ 0 (η; λ) = 1, the above two formulas reduce to the well-known Rodrigues formulas for the Laguerre and Jacobi polynomials, (E.6) and (E.19). For ℓ = 1 these two formulas are equivalent to the Rodrigues-type formulas (77) and (52) in Gomez-Ullate et al. [7]. For lower ℓ, it is straightforward to verify the shape invariance relation (4.2) by direct calculation. In [15] it was shown that the shape invariance relation is attributed to a new polynomial identity of degree 3ℓ involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities in [15].
It is a good challenge to derive more explicit expressions of the Rodrigues formulas for each of the four families of the exceptional orthogonal polynomials.

Forward and backward shift relations
Again it is straightforward to verify that the above Rodrigues formulas provide the explicit forms of the four families of the X ℓ polynomials (2.1)-(2.4) for lower ℓ and n by direct calculation. For analytic proof, one needs to verify that the above forward (4.3) and backward (4.4) shift relations are actually satisfied by the four families of the X ℓ polynomials (2.1)-(2.4). The forward and backward shift relations are attributed to new polynomial identities of degree 2ℓ + n − 1 and 2ℓ + n involving cubic products of the Laguerre or Jacobi polynomials in a similar way to the shape invariance relations [15]. The forward (F) (4.3) and backward (B) (4.4) shift relations read explicitly where P ℓ,n is given in (2.5).
The new polynomial identities are as follows: We do not show those for the J1 polynomials, since J1 and J2 are related by the mirror image, η ↔ −η and renaming of the coupling constants g ↔ h. Thus the corresponding identities are essentially the same. The derivative terms in the above identities (5.3)-(5.8) can be eliminated by using the forward shift relations (E.2) and (E.15) for the Laguerre and Jacobi polynomials. In Appendix B, the forward and backward shift relations for the four types of X ℓ polynomials are proven elementarily.

Invariant polynomial subspace
Another characterisation of polynomials satisfying differential equations is the existence of invariant polynomial subspaces [8]. For the Laguerre and Jacobi differential equations with H 0 (λ) (3.7), (3.8), the space V 0,n of degree n polynomials in η is invariant H 0 (λ)η n = E n (λ)η n + lower orders.

Integration formulas
Another well-known construction method of orthogonal polynomials is the Gram-Schmidt orthonormalisation of certain basis vectors under a given inner product specified by a weight function. Let us introduce two types of inner products * , * ℓ,λ and ( * , * ) λ : in which p(η) and q(η) are arbitrary functions and W ℓ (η; λ) is the weight function for the X ℓ polynomials, whereas W (η; λ) is the weight function for the Laguerre or Jacobi polynomials (2.11).
Here we present the integration formulas: where λ ′ = λ + ℓδ +δ. For a proof, see Appendix C.

Gram-Schmidt orthonormalisation
As for the direct application of Gram-Schmidt orthonormalisation, one orthonormalises {Ξ ℓ,λ [p n (η)]} in (6.8) with respect to the inner product * , * ℓ,λ . The following choice of the function p n is made: with which the inner products (p n (η), p m (η)) λ ′ are easily expressed in terms of gamma functions. This corresponds to the choice of the basis vectors in (6.6) and (6.7).
The Gram-Schmidt orthonormalisation procedure at the n-th step determines P ℓ,n (η; λ) by the following formula which is to be compared with the procedure for the undeformed (ℓ = 0) polynomial where λ ′ = λ + ℓδ +δ. These two orthonormalisations are essentially the same. In fact, by applying Ξ ℓ,λ [ · ] to (8.2) and using (2.8), we have Comparing these we obtain In other words, the Gram-Schmidt orthonormalisation for the undeformed (ℓ = 0) polynomials provides that for the X ℓ polynomials, too.

Generating functions
Generating functions for orthogonal polynomials have played another important role in classical analysis. Let us define generating functions for the X ℓ polynomials P ℓ,n and for the undeformed polynomials P n , The latter is quite well known where G (α) (t, x) and G (α,β) (t, x) are given in (E.8) and (E.21). Since P ℓ,n (η; λ) is expressed linearly in terms of P n (η; λ ′ ) (2.5) (λ ′ = λ + ℓδ +δ), the generating function G ℓ (t, η; λ) is expressed simply in terms of the known G(t, η; λ ′ ): t n d 0 (n, λ)P ℓ,n (η; λ) The forward shift relation (2.23) implies Here we present the concrete forms of the generating functions: where R def = 1 − 2ηt + t 2 . Next let us introduce the double generating function, that is the generating function of the generating functions G ℓ (t, η; λ): For L1 and L2 cases, the explicit forms are: which are obtained by using the two shifted generating functions (E.9) and (E.10) in Appendix E.1. It is a good challenge to derive the double generating functions for the X ℓ Jacobi polynomials.

Three term recurrence relations
Three term recurrence relations are one of the most fundamental characteristics of the ordinary orthogonal polynomials of one variable. Obviously the exceptional orthogonal polynomials do not satisfy these relations. Nevertheless, being deformations of ordinary orthogonal polynomials, the X ℓ polynomials are expected to retain certain reminiscent properties of the three term recurrence.

Zeros of X ℓ polynomials
The zeros of orthogonal polynomials have always attracted the interest of researchers. In the case of X ℓ polynomials P ℓ,n (η; λ), it has n zeros in the domain where the weight function is defined, that is (0, ∞) for the L1 and L2 polynomials and (−1, 1) for the J1 and J2 polynomials. See for example, Section 5.4 of [19]. The behaviour of these zeros are the same as those of other ordinary orthogonal polynomials. This is guaranteed by the oscillation theorem of the one-dimensional quantum mechanics, since P ℓ,n (η; λ) are obtained as the polynomial part of the eigenfunctions of a shape invariant quantum mechanical problem [1,2].
Here we discuss the location of the extra ℓ zeros of the exceptional orthogonal polynomials, which lie in various different positions for the different types of polynomials. So far we have verified by direct calculation for lower ℓ and n: The ℓ extra zeros of L1 polynomials are on the negative real line (−∞, 0). Those of the L2 X ℓ:odd polynomials are 1 real negative zero which lies to the left of the remaining 1 2 (ℓ − 1) pairs of complex conjugate roots. The L2 X ℓ:even polynomials have 1 2 ℓ pairs of complex conjugate roots. For L2, these ℓ additional roots lie to the left of the n real zeros.
The situations for the X ℓ Jacobi polynomials are a bit more complicated. The J1 X ℓ:odd polynomials have 1 real negative root which lies to the left of the remaining 1 2 (ℓ − 1) pairs of complex conjugate roots with negative real parts. The J1 X ℓ:even polynomials have 1 2 ℓ pairs of complex conjugate roots with negative real parts. For J1, some of the complex roots have real parts between −1 and 0. The J2 X ℓ:odd polynomials have 1 real positive root which lies to the right of the remaining 1 2 (ℓ − 1) pairs of complex conjugate roots with positive real parts. The J2 X ℓ:even polynomials have 1 2 ℓ pairs of complex conjugate roots with positive real parts. For J2, some of the complex roots have real parts between 0 and 1.

Summary and comments
We have given an in-depth study of the properties of the exceptional (X ℓ ) polynomials discovered recently in [1,2,15]. Our main focus is the derivation of certain equivalent but much simpler looking forms of the X ℓ polynomials. The derivation is based on the analysis of the second order differential equations for the X ℓ polynomials within the framework of the Fuchsian differential equations in the entire complex plane. These new forms of the X ℓ polynomials allow easy verification of the actions of the forward and backward shift operators on the X ℓ polynomials, and provide direct derivation of the Rodrigues formulas and the generating functions. The structure of the invariant polynomial subspaces under the Fuchsian differential operators is elucidated. The bases of the invariant polynomial subspaces provide a simple substitute of the three term recurrence relations. The Gram-Schmidt construction of the X ℓ polynomials starting from the above bases is demonstrated with the help of an integration formula. The properties of the extra zeros of the X ℓ polynomials are discussed. Some technical details are relegated to the Appendices. The proof of the equivalence of the new and original forms of the X ℓ polynomials is given. Simple proofs of the forward and backward shift operations are shown. The integration formula is elementarily proven. Various fundamental formulas of the Laguerre and Jacobi polynomials are supplied for easy reference.
Let us mention that the same method, deformation in terms of a degree ℓ eigenpolynomial, applied to the discrete quantum mechanical Hamiltonians for the Wilson and Askey-Wilson polynomials produced two sets of infinitely many shape invariant systems together with exceptional (X ℓ ) Wilson and Askey-Wilson polynomials (ℓ = 1, 2, . . .) [16]. It will be interesting to carry out the same analysis in these discrete cases.
Finally, concerning the issue of global solutions of Fuchsian differential equations, we would like to make a comment on the well-known theorem by Heine-Stieltjes [20]. It asserts the existence of a polynomial solution for the differential equation in which only two coefficient functions A(x) and B(x) are specified. They are degree p + 1 and p polynomials, respectively. In this case a polynomial C(x) of degree p − 1 is not given at the beginning but is determined so that the equation admits a degree n polynomial solution.
Thus the problem setting is not a proper Fuchsian differential equation and the process of determining C(x) and the polynomial solution is purely algebraic. We consider these polynomial solutions do not qualify to be 'global solutions' of the ordinary Fuchsian equations. After this paper was arXived, some of the results were re-derived in terms of the Darboux-Crum transformations [17,18].
We fix ℓ and use new parameters α and β instead of g and h, By using the forward shift relation for the Jacobi polynomial (E.15), the polynomials ξ ℓ and P ℓ,n are expressed as Here we provide the proof for the equivalence of the two expressions (A.1) and (A.2) step by step: × (n + α + 1)P (α,β) n (η) − (α + 1)P (α+1,β−1) n (η) The other exceptional polynomials, the J1 case is obtained from the above J2 case by (E.14). The equality of the exceptional Laguerre polynomials given in [1,2] and those given in this paper (2.1) will not be given here, since the L1 and L2 cases are obtained from the J1 and J2 cases by the limit (E.28).

B Forward and backward shift relations
Here we provide proofs for the forward (4.3) and backward (4.4) shift relations which apply equally for the four types of X ℓ polynomials. The method is elementary based on various identities of the Laguerre and Jacobi polynomials.

C Proof of the integration formula
Here we present a simple proof for the integration formulas (7.1) or (7.2). We evaluate where λ ′ = λ + ℓδ +δ. In (i) we have integrated by part and in (ii) we have used which is shown by using (2.18) and (2.20) to eliminate ξ ℓ (η; λ + δ).